A.H.Alizadeh.N.K.Nomikos Transportation Research Part B 41 (2007)126-143 129 Eq.(1)can be rearranged to represent the present value relationship,where the current ship price,P,is ex- pressed in terms of the expected price of the vessel,expected operational profits and expected rate of return,in the following expression P,= rE,P41+E,Ⅱ+ 1+E,R+1 (2) Eq.(2)is in fact a one period present value model;through recursive substitution and some algebraic manip- ulation,P,can be written as the sum of the present values of the future profits plus the terminal or resale value, Pof the asset.Mathematically +E,R+)厂 E,+H+ +E,R+） (3) Eq.(2)can also be written in logarithmic form;however,in this case it is not possible to perform recursive substitutions to write the log of price (InP,)in terms of the log of discounted expected earnings and log of discounted expected terminal value of the asset.Campbell and Shiller(1987)suggest a way round this by using a first-order Taylor series expansion and linearising(1)around the geometric mean of P and IT(P and IT)to give In(1+E,R+1)=pIn(E,P+)+(1-p)In(E:I+1)-In P:+k (4) where p=P/(P+IT)and k =-In(p)-(1 p)In(1/p-1).Letting Ep+1 In(E,P:+1),Ei+1=In(1 E,R+1) and E+=In(E,I,+1),Eq.(2)can be written as P,=pEP+1+(1-p)E元+1-E+1+k (5) which can be solved recursively forward to yield =∑l-pE4a-pPE4+prE+k1-p/0- (6) Since prices and operating profit series are non-stationary,Eq.(6)should be transformed in such a way so as to derive a model with stationary variables.Following Campbell and Shiller(1987),we use the cointegration relationship between the log-price and the log-earning series for such transformation;that is the log P/E ratio.This is done by subtracting n,from both sides of(6)which results in A-元= ∑pl-p）E4H--∑pE41H+Ep+kl-p)/I-p) (7) or Cp'(E,△元+1+H-E+1+i)+p(EpPn-E+)+k(1-p)/(1-p） (8) i-0 In the above setting p,-n,and p-n,are the log P/E ratio and log resale price-earning ratio,respectively. According to Campbell and Shiller(1987),the left hand side of Eq.(8)is the actual spread,and the right hand 4 It has been argued that many financial and economic time series are non-stationary.Such variables tend to have an increasing variance and do not show a tendency to revert to a long-run mean.In order to detect such behaviour in a variable one should use unit root tests such as the Phillips and Perron (1988)and Kwiatkowski et al.(1992).In general,it has been shown that correlation between non- stationary series does not accurately represent the true relationship between variables.However,there might be cases where two non- stationary variables can be related in the long-run through an equilibrium relationship,but deviate from such an equilibrium in the short run.Such a relationship is called a cointegrating relationship and implies that a linear combination of the two non-stationary series is stationary(Engle and Granger,1987).In our case for instance,although the log of ship prices and the log of earnings are non-stationary time series,their difference(i.e.the P/E ratio)should be stationary because ship prices and earnings are linked through the fundamental pricing relationship of Eq.(6).Thus,if the P/E ratio is too high or too low,we expect it to revert back to its long-run mean due to corrective movements in the level of earnings and ship prices.Eq. (1) can be rearranged to represent the present value relationship, where the current ship price, Pt, is ex￾pressed in terms of the expected price of the vessel, expected operational profits and expected rate of return, in the following expression Pt ¼ EtPtþ1 þ EtPtþ1 1 þ EtRtþ1 ð2Þ Eq. (2) is in fact a one period present value model; through recursive substitution and some algebraic manip￾ulation, Pt can be written as the sum of the present values of the future profits plus the terminal or resale value, Psc tþn of the asset. Mathematically Pt ¼ Xn i¼1 Yi j¼1 ð1 þ EtRtþjÞ 1 !EtPtþi þ Yn j¼1 ð1 þ EtRtþjÞ 1 !EtPsc tþn ð3Þ Eq. (2) can also be written in logarithmic form; however, in this case it is not possible to perform recursive substitutions to write the log of price (lnPt) in terms of the log of discounted expected earnings and log of discounted expected terminal value of the asset. Campbell and Shiller (1987) suggest a way round this by using a first-order Taylor series expansion and linearising (1) around the geometric mean of P and P (P and P) to give lnð1 þ EtRtþ1Þ ¼ q lnðEtPtþ1Þþð1 qÞlnðEtPtþ1Þ ln Pt þ k ð4Þ where q ¼ P=ðP þ PÞ and k = ln(q) (1 q)ln(1/q 1). Letting Etpt+1 = ln(EtPt+1), Etrt+1 = ln(1 + EtR+1) and Etpt+1 = ln(EtPt+1), Eq. (2) can be written as pt ¼ qEptþ1 þ ð1 qÞEptþ1 Ertþ1 þ k ð5Þ which can be solved recursively forward to yield pt ¼ Xn1 i¼0 qi ð1 qÞEtptþ1þi Xn1 i¼0 qi Etrtþ1þi þ qn Etpsc tþn þ kð1 qn Þ=ð1 qÞ ð6Þ Since prices and operating profit series are non-stationary, Eq. (6) should be transformed in such a way so as to derive a model with stationary variables. Following Campbell and Shiller (1987), we use the cointegration relationship between the log-price and the log-earning series for such transformation; that is the log P/E ratio.4 This is done by subtracting pt from both sides of (6) which results in pt pt ¼ Xn1 i¼0 qi ð1 qÞEtptþ1þi pt Xn1 i¼0 qi Etrtþ1þi þ qn Etpsc tþn þ kð1 qn Þ=ð1 qÞ ð7Þ or pt pt ¼ Xn1 i¼0 qi ðEtDptþ1þi Etrtþ1þiÞ þ qn ðEtpsc tþn EtptþnÞ þ kð1 qn Þ=ð1 qÞ ð8Þ In the above setting pt pt and psc t pt are the log P/E ratio and log resale price–earning ratio, respectively. According to Campbell and Shiller (1987), the left hand side of Eq. (8) is the actual spread, and the right hand 4 It has been argued that many financial and economic time series are non-stationary. Such variables tend to have an increasing variance and do not show a tendency to revert to a long-run mean. In order to detect such behaviour in a variable one should use unit root tests such as the Phillips and Perron (1988) and Kwiatkowski et al. (1992). In general, it has been shown that correlation between non￾stationary series does not accurately represent the true relationship between variables. However, there might be cases where two non￾stationary variables can be related in the long-run through an equilibrium relationship, but deviate from such an equilibrium in the short run. Such a relationship is called a cointegrating relationship and implies that a linear combination of the two non-stationary series is stationary (Engle and Granger, 1987). In our case for instance, although the log of ship prices and the log of earnings are non-stationary time series, their difference (i.e. the P/E ratio) should be stationary because ship prices and earnings are linked through the fundamental pricing relationship of Eq. (6). Thus, if the P/E ratio is too high or too low, we expect it to revert back to its long-run mean due to corrective movements in the level of earnings and ship prices. A.H. Alizadeh, N.K. Nomikos / Transportation Research Part B 41 (2007) 126–143 129